Problem 55 - Prealgebra

Question

Use the distributive property to combine similar terms. \(4 x-9 x\)

Step-by-Step Solution

Answer

The expression simplifies to \(-5x\).

Step 1: Identify the common factor

In the expression \(4x - 9x\), both terms have the variable \(x\). They can be combined using the distributive property because they are like terms.

Step 2: Use the distributive property

The distributive property allows us to factor out the common variable \(x\) from both terms: \(4x - 9x = (4 - 9)x\).

Step 3: Simplify the expression

Calculate \((4 - 9)\): \((4 - 9)x = (-5)x.\)So, the simplified expression is \(-5x\).

Key Concepts

Understanding Like Terms

When dealing with algebraic expressions, identifying "like terms" is crucial. Like terms are terms within an expression that have the same variable component. For example, in expressions like \(4x\) and \(-9x\), both terms have the variable \(x\).
This means they can be combined using arithmetic operations. By recognizing like terms, you can simplify expressions more easily, as it sets the stage for combining them efficiently. This understanding is foundational when working with algebraic equations that contain multiple variables and coefficients, as it allows us to focus on similar components for mathematical operations.

Combining Like Terms

Once you've identified like terms, the next step is to combine them. Combining like terms means performing arithmetic on their coefficients, the numerical values that stand in front of the variables. For example, in the expression \(4x - 9x\), because both terms contain the same variable \(x\), you can subtract the coefficients: \(4 - 9\).
When combining like terms:

• Keep the variable part unchanged.
• Perform the arithmetic operation on the coefficients only.
• Remember to maintain the sign of the coefficients during calculations.

Combining terms helps organize expressions, making them easier to solve or manipulate for further operations.

Simplifying Expressions

Simplifying an expression involves rewriting it in its simplest form. This often means reducing it down to the fewest possible terms.
Using distributive property is an effective strategy when simplifying expressions with like terms. In the example \(4x - 9x\), after identifying that both terms include the variable \(x\), you can "factor out" this common variable and simplify: \((4 - 9)x\).
Simplifying further, perform the calculation inside the parenthesis: \(-5x\).
The simplified expression is much cleaner and easier to work with. Simplifying not only makes expressions more manageable but also assists in solving equations and understanding their structure better.